Digital Signal Processing Reference
In-Depth Information
dealing for example with electrophysiological signals, they have in common their
low amplitude levels, and because of their frequency content, they can be severely
perturbed by power line noise. Some examples of such signals are electrocardiogram
(ECG), electromyogram (EMG), electrooculogram (EOG), electroencephalogram
(EEG) and extracellular recordings of neuronal activity. Removing this interference
might be essential to improve the SNR before starting any analysis on the signal,
especially if the effect of interest is nearby the noise frequency.
PLI originates from many sources, including the long wires between the sub-
ject and the amplifier, the separation between the measurement points (electrodes),
capacitive coupling between the subject (a volume conductor) and power lines, and
the low amplitude of the desired signals [
5
].
A common practice to remove PLI is to use a single fixed notch filter centered at
the nominal line frequency. If we use a second order FIR filter, the resulting band-
width might be too large, leading to signal distortion and possibly the elimination
of important content (for example, for diagnostic purpose). With a second order IIR
filter, we can control the bandwidth with the location of the poles. However, line fre-
quency, amplitude and phase might not be constant. If 5% variations in the frequency
of the power supply can take place, a wide enough notch filter would be required to
remove all that frequency band, so signal distortion is again an issue. Conversely, the
center of a very narrow notch filter may not fall exactly over the interference, failing
to remove it. Adaptive filters using a reference from the power outlet can track the
noise fluctuations, becoming perfect candidates for this application. As in PLI the
interference is sinusoidal, the noises in the ANC can be modeled as:
v
0
(
n
)
=
A
0
cos
(ω
0
n
+
φ
0
)
v
1
(
n
)
=
A
cos
(ω
0
n
+
φ).
(4.32)
where the phases
φ
0
and
φ
are fixed randomvariables distributed uniformly in [0
,
2
π)
.
If all signals are sampled at
f
s
Hz, the normalized angular frequency
ω
0
is equal to
2
f
s
, with
f
0
being the frequency of the sinusoid in Hz. As the frequency of the
reference signal would presumably be the same as the one from the noise
v
0
(
π
f
0
/
in
the primary input, the objective of the adaptive filter is to adjust its amplitude and
phase in order to match the noise
v
0
(
n
)
. To do this, it needs at least two coefficients.
However, in practice, the chance of having other forms of uncorrelated noise (not
necessarily sinusoidal) in the primary and reference inputs, would require the use of
further coefficients.
In [
6
] it was shown that the transfer function of the ANC, from
d
n
)
(
n
)
to
e
(
n
)
,using
an LMS algorithm can be approximated by
z
2
LA
2
4
E
(
z
)
−
2
z
cos
(ω
0
)
+
1
α
=
μ
H
(
z
)
=
)
=
α)
,
with
.
(4.33)
D
(
z
z
2
−
2
(
1
−
α)
z
cos
(ω
0
)
+
(
1
−
2
e
±
j
ω
0
.If
The zeros are located on the unit circle at
z
=
α
1, then
2
2
(
1
−
α)
=
1
−
2
α
+
α
≈
1
−
2
α.
